How to find an irrational number in <mrow

Pitrellais 2022-05-24 Answered
How to find an irrational number in Q [ 0 , 1 ]?
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Answers (1)

Kaylyn Ewing
Answered 2022-05-25 Author has 9 answers
For an "explicit" construction, consider z = i = 1 2 k i , where given k 1 < < k n , if i = 1 n 2 k i = a m ( n ) . Note that a m ( n ) < z < a m ( n ) + 2 1 k n + 1 < a m ( n ) + 2 2 m ( n ) so z∈Vm(n). Since m ( n ) , z is in all the V i . Its base-2 expansion contains infinitely many 1's but also has arbitrarily large gaps between the 1's, so can't be eventually periodic, and thus z must be irrational.
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